Which of the following equations describes a plane passing through the point $P = (3, -2, -1)$ and parallel to the plane $x - y - z + 3 = 0$?

Question

BlackTom AI · Accepted Answer

The task asks for a plane that passes through P = (3, -2, -1) and is parallel to the given plane x - y - z + 3 = 0. 
First, note that parallel planes share the same normal vector. The normal vector to x - y - z + 3 = 0 is n = (1, -1, -1). 
A plane with this same normal can be written in the form x - y - z + D = 0, where D is a constant to be determined.
To find D, substitute the coordinates of P into the equation: 3 - (-2) - (-1) + D = 0. 
Compute the terms: 3 + 2 + 1 + D = 0, which simplifies to 6 + D = 0, so D = -6. 
Therefore, the plane through P with the same normal is x - y - z - 6 = 0.
If there were alternative options, you would compare their normals: any correct option must have the same normal (1, -1, -1); any deviation in the normal would mean the plane is not parallel to the given plane. If any candidate had a different D but kept the same normal, it would correspond to a plane parallel to the given one but passing through a different point, which would not satisfy P. Conversely, a change in the normal vector would indicate a plane not parallel to the original. In short, enforcing both the parallelism (same normal) and the passage through P yields the specific equation x - y - z - 6 = 0.

Which of the following equations describes a plane passing through the point \(P = (3, -2, -1)\) and parallel to the plane \(x - y - z + 3 = 0\)?单项选择题

类似问题

Which of the following is the equation of the plane containing points 𝐴 = ( 0 , 0 , 0 ) , 𝐵 = ( 1 , 1 , 0 ) , and 𝐶 = ( 0 , 2 , − 1 ) .

Which of the following equations describes a plane passing through the point \(P = (3, -2, -1)\) and parallel to the plane \(x - y - z + 3 = 0\)?

Determine the scalar equation of the plane passing through the point (1,2,−1) with the normal vector (2,3,−4)? (I:1)

Given three points A(1,2,3), B(4,5,6) and C(7,8,9), which of the following is the correct scalar equation of the plane passing through these points? (K:1)

Which of the following equations describes a plane passing through the point \(P = (-1, 5, 4)\) and parallel to the plane \(2 x + y + 3 z - 13 = 0\)?

Determine the scalar equation of the plane passing through the point (1,2,−1) with the normal vector (2,3,−4)? (I:1)

Let ℓ be the line in R3 with vector equation x=[2,3,3]+t[−2,1,−2], where t∈R. Let P be the plane in R3 which contains the line ℓ and the point (1,1,1). Which one of the following is a general equation for the plane P?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单